Problem
ALG-B2-M04-P019 General power form
#19
★★★★★ Level 5 of 5
Let \(x_1,\ldots,x_n\ge0\), and let \(m\) be a positive integer. Prove \[\sum_{i=1}^n x_i^{m+1}\ge\frac1n\left(\sum_{i=1}^n x_i^m\right)\left(\sum_{i=1}^n x_i\right).\]
Hint. Order the \(x_i\). Then \(x_i^m\) has the same order.
Reorder the numbers so that \(x_1\le\cdots\le x_n\). Then \(x_1^m\le\cdots\le x_n^m\). Chebyshev gives \[\frac1n\sum x_i^{m+1}\ge\left(\frac1n\sum x_i^m\right)\left(\frac1n\sum x_i\right).\] Multiplying by \(n\) gives the result.
This compact generalization is useful for later modules on power means.